Method for the high-resolution evaluation of signals for one or two-dimensional directional or frequency estimation

ABSTRACT

A method is specified for the high-resolution evaluation and, if appropriate, for the recovery of narrow-band signals, received by a centrosymmetrical sensor group (AG), for directional estimation. A method is also specified for the high-resolution evaluation of superimposed non-attenuated harmonic signals for spectral analysis in the case, if appropriate, of multi-channel observations. For directional estimation, signals which are noisy and disturbed by multipath propagation can be prepared for recovery. The method for directional estimation is particularly suitable for signal evaluation in mobile radio systems or wireless communication systems, in high-resolution radar image processing systems, sonar systems, and medical technology systems. The method for frequency estimation is suitable for image processing applications, and for a multiplicity of spectral analysis applications.

BACKGROUND OF THE INVENTION

Method for the high-resolution evaluation of signals for one- ortwo-dimensional directional or frequency estimation.

The invention relates to a method for the high-resolution evaluationand, if appropriate, for the recovery of received narrow-band signalsfor one- or two-dimensional directional estimation. The invention alsorelates to a method for the high-resolution evaluation of superimposed,non-attenuated harmonic signals for one- or two-dimensional frequencyestimation in the case, if appropriate, of multi-channel observations.

One- or two-dimensional methods for evaluating the direction ofincidence of the different signals are known from R. Roy and T. Kailath,"ESPRIT--Estimation of signal parameters via rotational invariancetechniques", IEEE Trans. Acoust., Speech, Signal Processing, vol.ASSP-37, pp. 984-995, July 1989 and A. L. Swindlehurst and T. Kailath,"Azimuth/elevation direction finding using regular array geometrics",IEEE Trans. Aerospace and Electronic Systems, Vol. 29, pp. 145-156,January 1993.

Methods for one-dimensional or two-dimensional frequency estimation inthe case of single-channel observations are known from R. Roy, A.Paulraj and T. Kailath, ESPRIT--A subspace rotation approach toestimation of parameters of cisoids in noise", IEEE Trans. Acoustics,Speech, Signal Processing, Vol. ASSP-34, p.1340-1342, October 1986 andM. P. Pepsin and M. P. Clark, "On the performance of several 2-Dharmonic retrieval techniques" in Proc. 28th Asilomar Conference onSignals, Systems and Computers", Pacific Grove, Calif., November 1994.

Because of their simplicity and their power of resolution, the methods,known as ESPRIT methods, for signal parameter estimation by techniquesbased on displacement invariances (denoted below as Standard ESPRITmethods) are suitable for directional or frequency estimation. In thecase of directional estimation, displacement invariances signify thegeometrical displacement of identical sensor subgroups, and in the caseof frequency estimation the time-referred displacement of theequidistant sample value divided into subgroups. However, complexcalculations involving a relatively high outlay on computation aregenerally required in the Standard ESPRIT method. Again, with risingcorrelation between the signals, the standard ESPRIT method loses inaccuracy and cannot cope in the case of coherent signals. In the case oftwo-dimensional directional evaluation, all known methods for thehigh-resolution directionally sensitive evaluation of signals requireoptimization or search strategies which are expensive in terms ofcomputing time for the purpose of assigning the signals the spatialcoordinates determined in the two dimensions. Furthermore, thereliability of the known methods cannot be estimated, and thus noautomatic improvement in the recording of measured values can beinstituted in the presence of unsatisfactory reliability.

Directional evaluation is opening up a new field of application foritself with mobile radio or methods resembling mobile radio. When beingpropagated in a propagation medium, signals are subject to interferenceowing to noise. Owing to instances of diffraction and reflection, signalcomponents traverse different propagation paths and overlap one anotherat the receiver, and lead to their extinction effects. Furthermore,instances of overlapping of the signals occur in the case of a pluralityof signal sources. Among other methods, frequency division multiplex andtime division multiplex methods or a method known as code divisionmultiplex serve the purpose of facilitating the distinguishing of signalsources and thus of evaluating the signals.

SUMMARY OF THE INVENTION

It is the object of the invention to specify a method for thehigh-resolution evaluation of signals for one- or two-dimensionaldirectional or frequency estimation, which can be carried out withreduced computational outlay and an increased accuracy of resolution.This object is achieved by the inventive method for the high-resolutionevaluation of signals for one- or two-dimensional directional orfrequency estimation. The method according to the invention can bedenoted as Unitary ESPRIT (Estimation of signal parameters viarotational invariance techniques).

An important aspect of the invention is to be seen in that acentrosymmetrical data model is selected which is achieved by means of acentrosymmetrical sensor group for the directional estimation, andequidistant sampling in the case of the frequency estimation. Theproperty of centrosymmetrical sensor groups with an invariancestructure, namely that they have a centrosymmetrical system matrix andthe phase factors of the directions of incidence of the signals arelocated on the unit circle, is utilized in the directional estimation.Calculation can be performed using predominantly real values owing tothe centrosymmetrical data model.

The method can be used in one or two evaluation dimensions. In the caseof the directional estimation of the incident wavefronts, the evaluationdimensions relate to the one or two angles of the direction ofincidence. In the case of the frequency estimation, the evaluationdimensions relate to the dimensions in which the frequency is to beestimated (1 time and 1 spatial dimension or 2 spatial dimensions). Analgorithm for carrying out the evaluation method can be specified bothfor one-dimensional and for two-dimensional sensor arrangements inclosed form. Search or optimization tasks with a high outlay oncomputation, which are usually required for calculations in the case oftwo-dimensional sensor arrangements, are avoided. By comparison with theStandard ESPRIT, only half the sample values are required for the sameaccuracy in the evaluation of uncorrelated signals or signal components.Said advantages of the closed algorithm lead to saving in computing timeand thus to a better suitability of the method according to theinvention for real time tasks.

In the case of the one-dimensional method, a purely real calculation anda reliability test of the solutions determined are possible. In the caseof the two-dimensional method, there is the advantage of obtaining aclosed solution rule which was rendered possible by an automaticpairing--of the single complex calculation--of the solutions determinedin the two dimensions.

The accuracy of the determination of the direction of incidence formutually correlated signals is improved by the method according to theinvention.

BRIEF DESCRIPTION OF THE DRAWINGS

The features of the present invention which are believed to be novel,are set forth with particularity in the appended claims. The invention,together with further objects and advantages, may best be understood byreference to the following description taken in conjunction with theaccompanying drawing, and in which:

FIG. 1 shows an evaluation device for carrying out the method accordingto the invention for the high-resolution evaluation of narrow-bandsignals in a mobile radio scenario with multipath propagation;

FIGS. 2a-2e shows in a to e one- and two-dimensional centrosymmetricalsensor groups for receiving narrow-band signals, as well as examples ofsubgroup formation;

FIGS. 3a and 3b shows in a a one-dimensional, and in b a two-dimensionalsensor group with incident wavefronts of different signals or signalcomponents and the associated angles of incidence;

FIGS. 4a and 4b shows a comparison of the evaluation results of theStandard ESPRIT (4a) and the Unitary ESPRIT method according to theinvention (4b) for three mutually correlated signals in a representationof the phase factors on the unit circle after 80 test runs, theinadmissible solutions of the Unitary ESPRIT method being provided witha mark ⊕;

FIGS. 5a and 5b specifies, in a diagrammatic representation, a receivedsignal together with subsequent harmonic analysis for the methodaccording to the invention for frequency estimation; and

FIG. 6 is a table showing the condition and advantages of the method ofthe present invention.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

Conditions and advantageous effects of the method according to theinvention are specified in FIG. 6.

The method according to the invention can be applied for the purpose ofobtaining from received narrow-band signals information on the directionof the incident wavefronts of the signals; this can also concern asingle signal with different signal components, subject to interferenceor also free from interference. The method according to the inventioncan also be used to obtain a composition of the incident signals interms of harmonic frequency components.

The method for directional estimation is explained below by way ofexample with the aid of FIGS. 1 to 4b.

The sensor group comprises M elements. A one-dimensional sensor group ora planar two-dimensional sensor group can be involved. The sensor groupselected in the exemplary embodiment is a uniform linear antenna groupAG with an element spacing of less than or equal to half the wavelengthλ. As may be seen from FIGS. 3a and 3b, the wavefronts of the signals orsignal components impinge on the one-dimensional antenna group at anangle θ_(k) in each case. The window length N is selected in this casesuch that the angle of incidence θ_(k) can be regarded as constantduring the samplings of a window length N. The directional evaluation isbased on the circumstance that a signal arrives with a time delay at thedifferent antenna elements. Consequently, a phase shift which is afunction of the direction of incidence θ_(k) exists between the samplevalues of a signal at the different antenna elements. By ascertainingthe phase shift, it is possible to determine the direction of incidenceθ_(k) of the signal. In the case of two-dimensional sensor groups, thedirections of incidence are evaluated in terms of azimuth angle andelevation angle. The determination of direction is based on theassumption that all the signal components have the same carrierfrequency.

The configuration of a sensor group used for the method according to theinvention for directional estimation is, however, subject to acondition. The sensor group must be centrosymmetrical, that is to saythe geometrical arrangement of the elements must be symmetrical in pairswith reference to a center point, and the complex properties ofsymmetrical sensor elements must be identical (various forms of designare specified in FIGS. 2a-2c). In addition, a one-dimensional sensorgroup must exhibit invariance in the direction of a spatial coordinate,and a two-dimensional sensor group must have this property in twodirections which need not necessarily be orthogonal to one another. Thefollowing notation is used below: column vectors or matrices are markedby bold lower case or upper case letters; transposed, conjugate complexor adjoint matrices and vectors are given the supplement T, * or H.

The system matrix A of the sensor group is centrosymmetrical andtherefore fulfils specific conditions, which can be described by theequation (1)

    II.sub.M A*=AΛAεC.sup.M×d             (1)

the complex matrix Λ being a unitary diagonal matrix of dimension d×d,and d specifying independently of time over a window length N the numberof the dominant incident signal components. II_(M) is an antidiagonalpermutation matrix of dimension M. It may also be remarked that thesystem matrices of the two subgroups, which are to be formed, of thesensor group must likewise fulfil the condition according to equation(1).

It should be noted concerning the narrow-band nature of the signalsreceived by the centrosymmetrical sensor group that no noticeable changein the complex envelope of the signal components is allowed to occurduring the propagation of the wavefronts, belonging to the signalcomponents, along the antenna aperture. The narrow-band nature can alsobe dictated by narrow-band filtering of the received signals.

The number of the sampled values N can likewise be freely selected, theaccuracy of estimation rising with increasing number N of the sampledvalues, but as does the dimension of the measured value matrix Xdetermined by the number of the elements M and the number of the sampledvalues N, in which case x_(i) (k) with (i=1,2 . . . M) and (k=1,2 . . .N) specifies the kth sampled value of the ith sensor, and the measuredvalue matrix x has the form: ##EQU1##

The processing of matrices of high dimension is more complicated thanthat of matrices of smaller dimension. The same also holds for complexmatrices determined by real and imaginary parts, by comparison with realmatrices. A smaller outlay on processing for signal evaluation methodsis a precondition for the use of these methods in real time systems.

The method is based on available measured values received by the antennagroup AG and subsequently conditioned. The method is carried out by wayof example in FIG. 1 for the high-resolution evaluation of narrow-bandsignals for one-dimensional directional estimation in an evaluationdevice AE. Subsequently, the directions of incidence θ_(k) determinedcan be used to separate the signal components x_(k) for thereconstruction of the wavefronts, and the source signals s₁,s₂ can beoptimally reconstructed. It is also possible to detect the direction ofthe signal sources s₁,s₂. Another possibility is generation oftransmitted signals which takes account of the propagation conditions byevaluating the received signals.

In the area of use of the method according to the invention representedin FIG. 1, the signals s₁,s₂, for example in a mobile radio scenario(mobile units MT1, MT2) with multipath propagation are affected bydiffraction and refraction at obstacles H, and thus subjected tosplitting into different signal components x_(k). Signal componentsx_(k) of different signals s₁,s₂ arrive at the antenna group AG. Inaddition to an antenna element, each sensor includes a device forconverting the radio-frequency signal or the radio-frequency signalcomponent received by means of the antenna element into a complexbaseband signal, which is then sampled. The further evaluation,described below, of the complex baseband signals is performed in theevaluation device AE.

The first method step is to read the sampled values x_(i) (k) for eachantenna element in the same sequence into the measured value matrix X.Should only one sampled value x_(i) (k) be available, a spatialsmoothing of the sampled values x_(i) (k) must follow. An equivalentsmoothing would be necessary for a frequency estimation by means of onlyone sensor element. However, this smoothing can precede the method ineach case. Methods for spatial smoothing are known from B. Widrow etal., "Signal Cancellation Phenomena in Adaptive Antennas: Cause andCures", in IEEE Trans. on Antennas and Propagation, Vol. AP-30, pages469-278, May 1982 and S. C. Pei et al., "Modified Spatial Smoothing forCoherent Jammer Suppression without Signal Cancellation", in IEEE Trans.on Acoustics, Speech and Signal Processing", Vol. ASSP-36, pages412-414, March 1988. In the case of spatial smoothing, sensor groups aresubdivided into a plurality of subgroups and the sampled measured valuesare pre-averaged so that it is also possible simultaneously to detect anumber, corresponding to the number of the subgroups formed, of coherentsignals or signal components x_(k) when they are incident from differentdirections. In the method according to the invention, it is possibleafter spatial smoothing simultaneously to detect the number,corresponding to double the number of the subgroups formed, of coherentsignals or signal components x_(k).

After initialization, the complex measured-value matrix X is transformedinto a second, purely real matrix T(X) in accordance with equation (3):

    T(X)=Q.sub.M.sup.H  XII.sub.M XII.sub.N !Q.sub.2N          (3)

The matrices Q_(M) ^(H) and Q_(2N) are selected as unitary, left II-realmatrices which are selected in accordance with equations (4) and (5):##EQU2## (I_(n) is an n-dimensional unitary matrix, II_(n) is ann-dimensional antidiagonal permutation matrix).

A left II-real matrix generally satisfies the condition II_(p) Q*=Q withQεC^(p)×q. II_(p) is an antidiagonal permutation matrix with II_(p)εR^(p)×p.

This general form of equation (3) can thus be simplified to equation(6), ##EQU3## if the complex measured-value matrix X is additionallysplit up into two submatrices (Z₁,Z₂) of equal size: ##EQU4##

The middle row can be omitted for an even number of sensor elements M,otherwise z^(T) specifies the row vector of the central row.

The second, purely real matrix T(X) has the dimensions M×2N, and thusdoubles the available matrix elements simply by relatively uncomplicatedcomputing operations. Doubling dimension of the measured-value matrix Xproduces a forward/backward averaging of the measured values which isinherent to the method.

A signal subspace estimation is carried out as the following methodstep. Methods which can be used for this purpose are explained in moredetail in A. J. Van der Veen, E. F. Deprettere and A. L. Swindlehurst,"Subspace-based signal analysis using singular value decomposition",Proc. IEEE, Vol. 81, pages 1277-1308, September 1993. A signal subspacematrix T(X) with the dimension (M×d), whose d columns define thed-dimensional signal subspace, is attained from the second, purely realmatrix E_(s). If the number of the sampled values N does not correspondto the number d of the dominant signal components x_(k), a reduction inrank thereby takes place. The number d of the dominant signal componentsx_(k) can be known a priori, and thus be available for the method, or isdetermined in this method step. The determination of the d dominantsingular values or eigenvalues representing the dominant signalcomponents x_(k) can be performed by selecting the singular values oreigenvalues located above a threshold value represented by a large powerdifference. In many methods for estimating signal subspaces, thisdetermination is also included implicitly. A method known to the personskilled in the art as singular value decomposition of the secondarymatrix T(X) is selected for determining the signal subspace matrix E_(s)produced from the second, purely real matrix T(X). The eigenvaluedecomposition of the estimated covariance matrix T(X)T^(H) (X) or aSchur-like signal subspace estimation method can also be selected.

As an example of the one-dimensional method, the uniform linear antennagroup AG is divided into two, identical subgroups which are, however,shifted by an element spacing Δ; FIGS. 2a-2c shows various possibilitiesfor this. It is to be noted in this case that the subgroups aresymmetrical to one another in relation to the center point of thegroup--this can be the case only with antenna groups AG which arealready symmetrical. As a rule, as large as possible an overlap of thesubgroups is desired, since thereby each subgroup can have a maximumnumber m of antenna elements and as high as possible a resolution can beachieved. Thus, the subgroup formation in accordance with FIG. 2a isselected in the exemplary embodiment.

In the case of maximum overlap and given a constant element spacing, thespacing of the two subgroups Δ is equal to this element spacing Δ. Inaddition, in the exemplary embodiment of the method according to theinvention the centrosymmetrical property of the antenna group AG is,furthermore, limited to uniform, that is to say identical, antennaelements. In the case of the failure of individual antenna elements,uniform antenna groups AG can be more easily matched while retaining thesymmetry.

Selection matrices K₁,K₂ must be set up in order to set up a possiblyoverdetermined system of equations for the signal subspace matrix E_(S).These selection matrices K₁,K₂ are obtained from a similaritytransformation in accordance with equation (8) from centrohermitianmatrices.

    K.sub.1 =Q.sub.m.sup.H (J.sub.1 +J.sub.2)Q.sub.M K.sub.2 =Q.sub.m.sup.H (J.sub.1 +J.sub.2)Q.sub.M                                 (8)

Auxiliary matrices J₁,J₂ εR^(m)×M : ##EQU5## are yielded, for example,for the antenna group AG selected, for example, in accordance with FIG.2a (number of elements M=6, maximum overlap with number of subgroupelements m=5).

The auxiliary matrix J₁ selects the elements of the first subgroup, andthe auxiliary matrix J₂ selects the elements of the second subgroup. Theresult is the selection matrices K₁,K₂ in the case of a selection of theleft II-real matrices Q_(m) ^(H),Q_(M) in accordance with equations (4)and (5): ##EQU6##

It is now possible to set up a system of equations in accordance withequation (11):

    K.sub.1 E.sub.s Y≈K.sub.2 E.sub.s                  (11)

The again purely real solution matrix Y,YεR^(d)×d can be foundapproximately with the aid of known solution methods for systems ofequations such as the least squares method. A unique solution can bedetermined if the number of the subgroup elements m corresponds to thenumber of dominant signal vectors d. If the number of the subgroupelements is larger, the system of equations is overdetermined, and asolution which is optimum for the respectively selected solution methodis determined.

The determination of the eigenvalue matrix Ω from the solution matrix Yis carried out via an eigenvalue decomposition in accordance withequation (12):

    Y≈TΩT.sup.-1 εR.sup.d×d        (12)

The eigenvalue matrix ΩεR^(d)×d includes the eigenvalues ω_(k)(Ω=diag(ω_(k))) on its diagonal. The matrices T and T⁻¹ represent acolumn matrix of the eigenvectors or the same in an inverted form. Theeigenvalues can also be determined via a Schur decomposition.

A reliability test which particularly distinguishes the one-dimensionalmethod according to the invention tests all the eigenvalues ω_(k)determined for their properties. If only real eigenvalues ω_(k) areestablished, the eigenvalues ω_(k) determined can be regarded asreliable. In the case of the occurrence of conjugate complex solutions,this reliability does not obtain, and it is necessary to repeat themethod with a larger number of sensor elements M or a larger number ofsampled values N.

FIGS. 4a and 4b show a comparison of the evaluation results of theStandard ESPRIT (4a) and the Unitary ESPRIT method (4b) for threemutually correlated signals in a representation of the phase factorse^(j)μk on the unit circle after 80 test runs.

FIG. 4b shows the unit circle with the phase factors e^(j)μk determinedby means of the method according to the invention. In the case of areliable result, all the phase factors e^(j)μk lie on the unit circle,and the eigenvalues ω_(k) are therefore real. Three of the test runsrepresented in FIG. 4b did not yield a reliable result--conjugatecomplex solutions were obtained and the method must be carried out anewusing an improved database. In comparison thereto, FIG. 4a shows thephase factors, determined with a substantially lower accuracy, in thecase of the use of the Standard ESPRIT method.

The directions θ_(k) of incidence of the signal components x_(k) for thedirectional estimation of the signals s₁,s₂ to be evaluated aredetermined via the equation (13)

    μ.sub.k ≈2 arc tan ω.sub.k =2 π/λ·Δ sin θ.sub.k          (13)

The wavelength λ is equal for all the signals or signal components.

Finally, the components of the source signals S can be determined bymeans of the general equation (14),

    S=A.sup.+ X                                                (14)

a suitable pseudo-inverse A⁺ of the system matrix A specified inequation (15) for the exemplary embodiment being calculated, forexample, via equation (16).

The estimated system matrix A of the exemplary embodiment has the form(M=6, d=4): ##EQU7##

The equation for forming the pseudo-inverse of the estimated systemmatrix A is:

    A.sup.+ =(A.sup.H A).sup.-1 A.sup.H                        (16)

Equation (14) is simplified to

    S=(DT.sup.-1 E.sub.S.sup.H Q.sub.M.sup.H)X(17)

when the real signal subspace matrix E_(S) has only orthogonal columns.The components x_(k) of the source signals s₁,s₂ are recovered bymultiplying a pseudo-inverse DT⁻¹ E_(S) ^(H) Q_(M) ^(H) of the estimatedsystem matrix A,AεR^(M)×d, containing the phase factors e^(j)μk of theeigenvalues ω_(k) determined, by the measured-value matrix X. In thiscase, the diagonal matrix DεC^(d)×d represents an arbitrarily selecteddiagonal matrix of dimension d×d.

The estimated system matrix obtained from the case of reception by meansof the evaluation device AE case of transmission and receptionidentical. The signals s₁,s₂ to be transmitted can now be decomposedinto signal components x_(k) in one way and be radiated in variousdirections, determined by the reception, and in an appropriately delayedfashion, so that they overlap in terms of power at the receiver.

Two-dimensional directional estimation

The two-dimensional evaluation requires that some method steps becarried out in parallel for the two evaluation dimensions. In the caseof the now two-dimensional, centrosymmetrical sensor group, there is noprescribed sequence for reading in the sample values with respect to thearrangement of the sensors, but the system matrix A must satisfy theform prescribed in equation (1).

The conversion of the complex measured-value matrix X into a second,purely real matrix T(X) of dimension (M×2N), which contains exclusivelyreal values and can be assigned to the measured values, and a signalsubgroup estimation for determining the real signal subspace matrixE_(S) by processing the real, M×2N-dimensional matrix T(X) while takingaccount of the d dominant vectors defining the signal subspace arecarried out in a way similar to the one-dimensional method.

The subgroup formation into two mutually symmetrical subgroups, andestablishing in each case two selection matrices K.sub.μ1,K.sub.μ2 andK.sub.ν1,K.sub.ν2 are carried out separately for the two dimensions x,yof the sensor group, the index μ being assigned to the dimension x andthe index ν to the dimension y. The subgroup formation of the twodirections of extent need not be performed in accordance with the samepoints of view, that is to say Δ_(x) need not be equal to Δ_(y) (spacingof the subgroups in the x-direction and y-direction), and m_(y) can beselected independently of m_(x) (m_(x),m_(y) being the number of thesubgroup elements in the x-direction and y-direction).

Two systems of equations are set up:

    K.sub.μ1 E.sub.S Y.sub.μ ≈K.sub.μ2 E.sub.S K.sub.ν1 E.sub.S Y.sub.ν ≈K.sub.ν2 E.sub.S           (18)

The solution matrices Y.sub.μ,Y.sub.ν are determined, for example, inturn by the least squares method.

Subsequently, the eigenvalues of the complex matrix Y.sub.μ +jY.sub.νare determined in accordance with equation (19).

    Y.sub.μ +jY.sub.ν =TΛT.sup.-1                 (19)

The complex eigenvalue matrix Λ contains the complex eigenvalues λ_(k)=(ω.sub.μk +jω.sub.νk) on its diagonal Λ=diag(λ_(k)). This means anautomatic pairing of the eigenvalues in x-direction (ω.sub.μk) andy-direction (ω.sub.νk).

The complex eigenvalues λ_(k) are evaluated in accordance with theazimuth angle θ_(k) and elevation angle φ_(k) using equations (20) to(22).

    ω.sub.μk =tan (μ.sub.k /2)ω.sub.νk =tan (ν.sub.k /2)(20)

    μ.sub.k =cos φ.sub.k sin θ.sub.k ν.sub.k =sin φ.sub.k sin θ.sub.k                                         (21)

    μ.sub.k =2 π/λ·Δ.sub.x μ.sub.k ν.sub.k =2 π/λ·Δ.sub.y ν.sub.k           (22)

Both angles represent the direction of incidence of the respectivesignal or the signal component, for which see FIG. 3b.

The applicability of the method according to the invention fordirectional estimation is not limited to a mobile radio environment, butcomprises in a similar way problems in radar or sonar technology,astronomy, the surveying of mobile radio channels or other problems inseismic or medical signal processing. Methods for the directionallysensitive evaluation of received signals, that is to say spatialfiltering, can be applied to the reception of electromagnetic, acousticand other types of waveform.

Frequency estimation

The method according to the invention can also be used to estimatefrequency components within a received signal, that is to say forspectral analysis. The dimension of the sensor group can be freelyselected in the case of frequency estimation. The arrangement of thesensor elements is not subjected to any limitations. The dimensions forthe sensor elements and the equidistant sample values must beinterchanged accordingly. The subgroup formation takes place with theaid of a subdivision, for example, of the sample values along the timeaxis. The high-resolution evaluation can be performed in aone-dimensional or two-dimensional fashion and is based overwhelminglyon real calculations. Only non-attenuated oscillations can be evaluated.

The dominant frequencies μ_(k) of the frequency mixture are determinedin accordance with the method steps outlined for the directionalestimation. FIG. 5a shows a diagrammatic view of a received signal s(t),represented at left, together with the result of the subsequent harmonicanalysis (represented at right) for the frequency estimation. Areliability test is possible for one-dimensional evaluation. The numberof the sensor elements can be reduced to one element. Radar andastronomical applications make use of spectral analysis, for example.Image processing may be named for two-dimensional frequency estimation(FIG. 5b), in which case an image is evaluated in the horizontal andvertical directions in accordance with the dominant frequencies, and thesensor elements, for example, be assigned to the individual pixels. Asingle sampled value detects a still image, while a plurality of samplevalues detect moving images.

The invention is not limited to the particular details of the methoddepicted and other modifications and applications are contemplated.Certain other changes may be made in the above described method withoutdeparting from the true spirit and scope of the invention hereininvolved. It is intended, therefore, that the subject matter in theabove depiction shall be interpreted as illustrative and not in alimiting sense.

What is claimed is:
 1. A method for high-resolution evaluation ofsignals for one-dimensional or two-dimensional directional estimation ina device for digital signal processing to which is assigned acentrosymmetrical sensor group having a number of sensors, comprisingthe steps of:storing sample values currently measured in the sensors ina complex measured-value matrix (x) having a dimension (M×N) determinedby the number of the sensors and a number of sampling instances;combining an M-dimensional antidiagonal permutation matrix (II_(M)) withthe conjugate complex measured-valued matrix (x*) and an N-dimensionalantidiagonal permutation matrix (II_(N)) to form a first matrix (II_(M)x*II_(N)); determining a centrohermitian matrix (x II_(M) x*II_(N)) fromthe complex measured-value matrix (x) and the first matrix (IIx*II_(N)); determining via an M-dimensional left II-real, adjoint matrix(Q_(M) ^(H)) and a 2N-dimensional left II-real matrix (Q_(2N)), startingfrom the centrohermitian matrix (x II_(M) x*II_(N)), a second, purelyreal matrix (T(x)) of a form T(x)=Q_(M) ^(H) (x II_(M) x*II_(N))Q_(2N)which contains exclusively real values, which can be assigned to themeasured values and --which has double the number of elements;undertaking by processing the second, purely real matrix (T(x)), asignal subspace estimation to determine a real signal subspace matrix(E_(S)) with d dominant vectors of the second, purely real matrix(T(x)); undertaking a subgroup formation, undertaken separately for eachevaluation dimension x,y of the method, of the centrosymmetrical sensorgroup into two subgroups displaced relative to one another, anddetermining two selection matrices (K₁,K₂,K.sub.μ1,2,K.sub.ν1,2) foreach evaluation dimension x,y in accordance with configuration of thesubgroups; calculating a solution, undertaken separately for eachevaluation dimension x,y of the method, of system equations prescribedby the signal subspace matrix(E_(S)), produced from the signal subspaceestimation, and the selection matrices (K₁,K₂,K.sub.μ1,2,K.sub.ν1,2) sothat in each case a solution matrix (Y,Y.sub.μ,Y.sub.ν) is available inaccordance with K₁ E_(S) Y=K₂ E_(S) ; determining an eigenvalue matrix(Ω,Λ) from the solution matrix (Y,Yμ,Yν), depending on the dimension x,yof the method; and the eigenvalues (ω_(k),λ_(k)) determined from theeigenvalue matrix (Ω,Λ) representing directional or frequency estimates.2. The method as claimed in claim 1, wherein a format conversion of theeigenvalues (ω_(k),λ_(k)) is carried out, and wherein for directionalestimation of directions of incidence (θ_(k) φ,_(k)) of signals to beevaluated, a determination, by undertaking separately for each directionof extent, representing the evaluation dimension x,y of the method, ofthe sensor group having M elements, of the directions of incidence(θ_(k) φ,_(k)) of the signals sampled with a window length of N measuredvalues is undertaken from the eigenvalues (ω_(k),λ_(k)) of theeigenvalue matrix (Ω,Λ).
 3. The method as claimed in claim 1, wherein adetermination of the d dominant singular values or eigenvaluesrepresenting dominant signal components (x_(k)) is provided by selectingsingular values or eigenvalues located above a special value representedby a large power difference.
 4. The method as claimed in claim 1,wherein a reconstruction of signal components (x_(k)) is undertaken bymultiplying a pseudo-inverse of an estimated system matrix (A)containing phase factors (e^(j)μk,e^(j)μν) of the eigenvalues determinedby the complex measured-value matrix (x).
 5. The method as claimed inclaim 1, wherein the method further comprises using self-decompositionof an estimated covariance matrix (T(x)T(x)^(H)) for determining thesignal subspace matrix (E_(S)) produced from the second, purely realmatrix (T(x)).
 6. The method as claimed in claim 1, wherein the methodfurther comprises using singular-value decomposition of the second,purely real matrix (T(x)) for determining the signal subspace matrix(E_(S)) produced from the second, purely real matrix (T(x)).
 7. Themethod as claimed in claim 1, wherein the method further comprises usinga Schur-like signal subspace estimation method for determining thesignal subspace matrix (E_(S)) produced from the second, purely realmatrix (T(x)).
 8. The method as claimed in claim 1, wherein thetwo-dimensional, planar, centrosymmetrical sensor group, which isinvariable in two directions, has M elements and the method has theevaluation dimensions x and y, and whereinthe selection matrices(K₁,K₂,K.sub.μ1,2,K.sub.ν1,2) are determined in accordance with the twodimensions x,y of the sensor group, the solution of the system ofequations, prescribed by the signal subspace matrix (E_(S)), producedfrom the signal subgroup formation, and the selection matrices (K₁,K₂)for the dimensions x and y is carried out in accordance withrelationships K.sub.μ,ν1 E_(S) Y.sub.μ,ν ≈K.sub.μ,ν2 E_(S) with a resultthat in each case solution matrix (Y.sub.μ,Y.sub.ν) in accordance withK₁ E_(S) Y≈K₂ E_(S) is available, assignment of the eigenvalues (λ_(k))of the solution matrices (Y.sub.μ,Y.sub.ν) is performed via a complexdetermination of the complex eigenvalue matrix (Λ) in accordance with arelationship Y.sub.μ +jY.sub.ν ≈TΛT⁻¹, and directions of incidencerepresented by azimuth (θ_(k)) and elevation angle (φ_(k)) aredetermined by relationships ω.sub.μk =tan (μ_(k) /2), ω.sub.νk =tan(ν_(k) /2); μ_(k) =cosφ_(k) sinθ_(k), ν_(k) =sinφ_(k) sinθ_(k) and μ_(k)=2 π/λ·Δ_(x) μ_(k), ν_(k) =2 π/λ·Δ_(y) ν_(k).
 9. The method as claimedin claim 1, wherein a spatial smoothing of measured values is undertakenbefore initialization of the complex measured-value matrix (x).
 10. Themethod as claimed in claim 1, wherein during the subgroup formation aslarge as possible an overlap of the subgroup elements takes place. 11.The method as claimed in claim 1, wherein the sensors are antennas forat least one of receiving and transmitting radio-frequencyelectromagnetic signals.
 12. The method as claimed in claim 11, whereinthe antennas are used in mobile radio systems.
 13. The method as claimedin claim 11, wherein the antennas are used in wireless communicationsystems.
 14. The method as claimed in claim 11, wherein the antennas areused in high-resolution radar image processing systems.
 15. The methodas claimed in claim 1, wherein the sensors are sound receivers fortransmitting and receiving acoustic signals.
 16. The method as claimedin claim 15, wherein the sensors are used in sonar systems.
 17. Themethod as claimed in claim 15, wherein the sensors are used in medicaltechnology systems.
 18. A method for high-resolution evaluation ofsignals for one-dimensional or two-dimensional frequency estimation in adevice for digital signal processing, for evaluating the signals inaccordance with a centrosymmetrical data model, comprising the stepsof:storing sample values currently measured in sensors in a complexmeasured-value matrix (x) having a dimension (M×N) determined by anumber of the sensors and a number of sampling instances; combining anM-dimensional antidiagonal permutation matrix (II_(M)) with theconjugate complex measured-valued matrix (x*) and an N-dimensionalantidiagonal permutation matrix (II_(N)) to form a first matrix (II_(M)x*II_(N)); determining a centrohermitian matrix (x II_(M) x*II_(N)) withthe complex measured-value matrix (x) and the first matrix (IIx*II_(N));determining via an M-dimensional left II-real, adjoint matrix (Q_(M)^(H)) and a 2N-dimensional left II-real matrix (Q_(2N)), starting fromthe centrohermitian matrix (x II_(M) x*II_(N)), a second, purely realmatrix (T(x)) of a form T(x)=Q_(M) ^(H) (x II_(M) x*II_(N))Q_(2N) whichcontains exclusively real values, which can be assigned to the measuredvalues and which has double the number of elements; undertaking a signalsubspace estimation to determine a real signal subspace matrix (E_(S))with d dominant vectors of the second, purely real matrix (T(x)) byprocessing the second, purely real matrix (T(x)); undertaking a subgroupformation, undertaken separately for each evaluation dimension x,y ofthe method, of the sensors into two subgroups displaced relative to oneanother, and determining two selection matrices(K₁,K₂,K.sub.μ1,2,K.sub.ν1,2) for each evaluation dimension x,y inaccordance with configuration of the subgroups; calculating a solution,undertaken separately for each evaluation dimension x,y of the method,of system equations prescribed by the signal subspace matrix (E_(S)),produced from the signal subspace estimation, and the selection matrices(K₁,K₂,K.sub.μ1,2,K.sub.ν1,2) so that in each case a solution matrix(Y,Y.sub.μ,Y.sub.ν) is available in accordance with K₁ E_(S) Y=K₂ E_(S); determining an eigenvalue matrix (Ω,Λ) from the solution matrix(Y,Yμ,Yν), depending on the dimension x,y of the method; and theeigenvalues (ω_(k),λ_(k)) determined from the eigenvalue matrix (Ω,Λ)representing directional or frequency estimates.
 19. The method asclaimed in claim 18, wherein a format conversion of the eigenvalues(ω_(k),λ_(k)) is carried out, and wherein a determination of theharmonic frequencies (μ_(k),ν_(k)) of the signals to be evaluated isundertaken for frequency estimation of signals which are sampled with awindow length of N measured values, a sensor group having M elements,and which are unattenuated during a window length.
 20. The method asclaimed in claim 18, wherein the one-dimensional sensor group has Melements, and the method has only one evaluation dimension x.
 21. Themethod as claimed in claim 20, wherein a reliability estimation isconducted by a test of the eigenvalues (ω_(k)) of the solution matrix(Y) for non-real solutions.
 22. The method as claimed in claim 18,wherein a determination of the d dominant singular values or eigenvaluesrepresenting dominant signal components (x_(k)) is provided by selectingsingular values or eigenvalues located above a special value representedby a large power difference.
 23. The method as claimed in claim 18,wherein a reconstruction of signal components (x_(k)) is undertaken bymultiplying a pseudo-inverse of an estimated system matrix (A)containing phase factors (e^(j)μk,e^(j)μν) of the eigenvalues determinedby the complex measured-value matrix (x).
 24. The method as claimed inclaim 18, wherein the method further comprises using self-decompositionof an estimated covariance matrix (T(x)T(x)^(H)) for determining thesignal subspace matrix (E_(S)) produced from the second, purely realmatrix (T(x)).
 25. The method as claimed in claim 18, wherein the methodfurther comprises using singular-value decomposition of the second,purely real matrix (T(x)) for determining the signal subspace matrix(E_(S)) produced from the second, purely real matrix (T(x)).
 26. Themethod as claimed in claim 18, wherein the method further comprisesusing a Schur-like signal subspace estimation method for determining thesignal subspace matrix (E_(S)) produced from the second, purely realmatrix (T(x)).
 27. The method as claimed in claim 18, wherein thetwo-dimensional, planar, centrosymmetrical sensor group, which isinvariable in two directions, has M elements and the method has theevaluation dimensions x and y, and wherein the selection matrices(K₁,K₂,K.sub.μ1,2,K.sub.ν1,2) are determined in accordance with the twodimensions x,y of the sensor group,the solution of the system ofequations, prescribed by the signal subspace matrix (E_(S)), producedfrom the signal subgroup formation, and the selection matrices (K₁,K₂)for the dimensions x and y is carried out in accordance withrelationships K.sub.μ,ν1 E_(S) Y.sub.μ,ν ≈K.sub.μ,ν2 E_(S) with a resultthat in each case solution matrix (Y.sub.μ,Y.sub.ν) in accordance withK₁ E_(S) Y≈K₂ E_(S) is available, assignment of the eigenvalues (λ_(k))of the solution matrices (Y.sub.μ,Y.sub.ν) is performed via a complexdetermination of the complex eigenvalue matrix (Λ) in accordance with arelationship Y.sub.μ +jY.sub.ν ≈TΛT⁻¹, and directions of incidencerepresented by azimuth (θ_(k)) and elevation angle (φ_(k)) aredetermined by relationships ω.sub.μk =tan (μ_(k) /2), ω.sub.νk =tan(ν_(k) /2); μ_(k) =cosφ_(k) sinθ_(k), ν_(k) =sinφ_(k) sinθ_(k) and μ_(k)=2 π/λ·Δ_(x) μ_(k), ν_(k) =2 π/λ·Δ_(y) ν_(k).
 28. The method as claimedin claim 18, wherein a spatial smoothing of measured values isundertaken before initialization of the complex measured-value matrix(x).
 29. The method as claimed in claim 18, wherein during the subgroupformation as large as possible an overlap of the subgroup elements takesplace.
 30. The method as claimed in claim 18, wherein the sensors areantennas for at least one of receiving and transmitting radio-frequencyelectromagnetic signals.
 31. The method as claimed in claim 30, whereinthe antennas are used in mobile radio systems.
 32. The method as claimedin claim 30, wherein the antennas are used in wireless communicationsystems.
 33. The method as claimed in claim 30, wherein the antennas areused in high-resolution radar image processing systems.
 34. The methodas claimed in claim 18, wherein the sensors are sound receivers fortransmitting and receiving acoustic signals.
 35. The method as claimedin claim 34, wherein the sensors are used in sonar systems.
 36. Themethod as claimed in claim 34, wherein the sensors are used in medicaltechnology systems.